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Maping a circle to a polygon by a polynomial!

Source: Iran 3rd round 2013 - Algebra Exam - Problem 5

September 11, 2013
algebrapolynomialtrigonometrycomplex numbersalgebra proposed

Problem Statement

Prove that there is no polynomial P∈C[x]P \in \mathbb C[x] such that set {P(z)  ∣  ∣z∣=1}\left \{ P(z) \; | \; \left | z \right | =1 \right \} in complex plane forms a polygon. In other words, a complex polynomial can't map the unit circle to a polygon. (30 points)
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